Regarding your concern about updates, yes the updates are automatically downloaded to your device however, it needs to be allowed by the user for download and installation. Other program updates, like the virus patterns are updated continuously once a new update is available.

Please note that the updates are not done via the Trend Micro account but it is only displaying which device/s are not yet updated.

If you are seeing on your Trend Micro account that the device needs to be updated, that means the device has failed to automatically update due to reasons like, device was not connected to the internet or the request to update was not allowed by the user.

It seems that you are referring to the upgrade notification that you received to install the latest version of Trend Micro.

Yes, the protection will be turned off because the installation process requires to uninstall the older version of Trend Micro. However, after restart, the installation process shall proceed. The process might take 15 to 30 minutes or more depending on the Internet and computer speed.

I run Trend Micro (version 7.0.2015) on the Mac OS X El
Capitan (version 10.11.6) side of my MacBook Pro, and Trend Micro [version 13.1.0
(43108)] on a virtual machine (Windows 7.1, sp1), using Parallels (version 13.2.0).
The virus pattern updates occur automatically on the Windows side, but not the
Mac side. I have been unable to find how to change either, but want very much
to automatically update the patterns on the Mac side as well. What am I
missing?

The Trend Micro program (on all platforms including Mac) is designed to check for updates every 3 hours, this update refers to pattern updates. If there is an available update, it will be downloaded and installed in the background. This feature is by design and is a vital part of the program itself. There may be chances that you are just missing the notifications that it is updating.